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Free calculus tool

Integral Calculator

Type a function, see it typeset as you go, and get the antiderivative with the steps.

Integral calculator

Integrand

Use ^ for powers, and sqrt( ), sin( ), ln( ), e^x and pi. Multiplication signs are optional: 2x sin(x) works.

xex\displaystyle xe^{x}

Try one

What you can type

The input reads ordinary calculator notation, the same as the derivative calculator. Multiplication signs are optional, brackets can be round, square or curly, and every function below accepts its argument with or without parentheses.

Accepted notation, the spellings that work, and how each one renders.
NotationType it asRenders as
Powersx^3x**3x³x3\displaystyle x^{3}
Implicit multiplication2x sin(x)2*x*sin(x)2xsin⁡(x)\displaystyle 2x\sin\left(x\right)
Division(x + 1)/(x - 1)x+1x−1\displaystyle \frac{x + 1}{x - 1}
Rootssqrt(x)cbrt(x)x^(1/2)x\displaystyle \sqrt{x}
Exponentialse^(2x)exp(2x)2^xe2x\displaystyle e^{2x}
Natural logarithmln(x)ln⁡(x)\displaystyle \ln\left(x\right)
Trigonometrysin(x)cos 2xtan(x)sec(x)csc(x)cot(x)sin⁡(x)\displaystyle \sin\left(x\right)
Powers of functionssin^2(x)sin(x)^2sin⁡2(x)\displaystyle \sin^{2}\left(x\right)
Inverse trigonometryarctan(x)atan(x)arcsin(x)arctan⁡(x)\displaystyle \arctan\left(x\right)
Absolute value|x|abs(x)∣x∣\displaystyle \left|x\right|
Constantspieπ\displaystyle \pi
Greek variablestheta^2sin(alpha)θ2\displaystyle \theta^{2}

Every answer carries + C. An indefinite integral is a whole family of functions, not one: any two antiderivatives of the same function differ by a constant, so the C is part of the answer rather than a decoration on it. For a second or third antiderivative there is a constant per integration, and the answer says so.

Any single letter is a variable, and so is a spelled-out Greek name like theta. When an expression contains more than one, choose which to integrate with respect to; the rest are treated as constants.

The techniques it uses

Integration is a search, not a procedure: unlike differentiating, there is no single rule that always applies, so the calculator tries techniques in the order a student would. Each step says which one it used.

Linearity

∫[af(x)+bg(x)]dx=a∫f(x) dx+b∫g(x) dx\displaystyle \int \left[a f(x) + b g(x)\right] dx = a\int f(x)\,dx + b\int g(x)\,dx

Term by term, with constants pulled out front. Tried first, because it turns one integral into several simpler ones.

Power rule

∫xn dx=xn+1n+1(n≠−1)\displaystyle \int x^{n}\,dx = \frac{x^{n+1}}{n+1} \quad (n \neq -1)

Any constant exponent except -1, where dividing by n + 1 would divide by zero. That one case gives a logarithm instead.

The n = -1 case

∫1x dx=ln⁡∣x∣\displaystyle \int \frac{1}{x}\,dx = \ln\left|x\right|

The absolute value matters: 1/x is defined for negative x too, and ln x alone would not be.

Substitution

∫f(g(x)) g′(x) dx=∫f(u) du\displaystyle \int f(g(x))\,g'(x)\,dx = \int f(u)\,du

The chain rule run backwards. Found by trying each subexpression as u and checking whether the integrand divided by u' comes out in terms of u alone.

Integration by parts

∫u dv=uv−∫v du\displaystyle \int u\,dv = uv - \int v\,du

For a product of unlike things. u is chosen by LIATE — logarithm, inverse trig, algebraic, trig, exponential — so what is left to integrate is simpler.

Partial fractions

1x2−1=1/2x−1−1/2x+1\displaystyle \frac{1}{x^{2}-1} = \frac{1/2}{x-1} - \frac{1/2}{x+1}

For a ratio of polynomials. The bottom is factored and the fraction split, which turns one hard integral into a sum of logarithms and arctangents.

Completing the square

∫dxx2+2x+5=ln⁡∣x+1+x2+2x+5∣\displaystyle \int \frac{dx}{\sqrt{x^{2}+2x+5}} = \ln\left|x + 1 + \sqrt{x^{2}+2x+5}\right|

Turns any quadratic under a root or a fraction into a standard form, which is what makes one rule cover a whole family.

Power reduction

∫sin⁡2x dx=x2−sin⁡2x4\displaystyle \int \sin^{2} x\,dx = \frac{x}{2} - \frac{\sin 2x}{4}

Double-angle and Pythagorean identities, for powers of sine, cosine and tangent that no substitution touches.

Not every function has an elementary antiderivative. e^(x^2), sin(x)/x and sqrt(1 + x^3) have none at all — that is a theorem, not a gap in this calculator — so when it says it cannot find one, that is sometimes the true answer. Differentiating always works; integrating does not.

Elementary antiderivatives

And the elementary antiderivatives it composes with them. Each one also works with a function of x inside, once substitution has dealt with the inner derivative:

  • ∫sin⁡x dx=−cos⁡x\displaystyle \int \sin x\,dx = -\cos x
  • ∫cos⁡x dx=sin⁡x\displaystyle \int \cos x\,dx = \sin x
  • ∫sec⁡2x dx=tan⁡x\displaystyle \int \sec^{2} x\,dx = \tan x
  • ∫sec⁡xtan⁡x dx=sec⁡x\displaystyle \int \sec x \tan x\,dx = \sec x
  • ∫tan⁡x dx=−ln⁡∣cos⁡x∣\displaystyle \int \tan x\,dx = -\ln\left|\cos x\right|
  • ∫sec⁡x dx=ln⁡∣sec⁡x+tan⁡x∣\displaystyle \int \sec x\,dx = \ln\left|\sec x + \tan x\right|
  • ∫ex dx=ex\displaystyle \int e^{x}\,dx = e^{x}
  • ∫ax dx=axln⁡a\displaystyle \int a^{x}\,dx = \frac{a^{x}}{\ln a}
  • ∫ln⁡x dx=xln⁡x−x\displaystyle \int \ln x\,dx = x\ln x - x
  • ∫dx1+x2=arctan⁡x\displaystyle \int \frac{dx}{1+x^{2}} = \arctan x
  • ∫dx1−x2=arcsin⁡x\displaystyle \int \frac{dx}{\sqrt{1-x^{2}}} = \arcsin x
  • ∫∣x∣ dx=x∣x∣2\displaystyle \int \left|x\right|\,dx = \frac{x\left|x\right|}{2}

Questions

Why is there always a + C?

Because an indefinite integral is a family of functions rather than one function. Differentiating kills a constant, so every antiderivative of the same integrand differs from the others only by one, and the C stands for all of them at once. Leaving it off is the single most common way to lose a mark on an integration question.

Does it show the steps?

Yes. Open "Show the steps" under any result and you get one line per technique, naming it and showing what the integral becomes. A substitution line says what u and du are, an integration by parts line says what u and dv are, and a partial fractions line shows the decomposition before it is integrated.

Why does it sometimes say it cannot do one?

Two different reasons. Some integrands have no elementary antiderivative at all — e to the x squared and sin x over x are the standard examples, and that is a theorem rather than a limitation. Others need a technique this calculator does not cover, such as trigonometric substitution or the Weierstrass substitution. Either way it says so instead of guessing.

How do I know the answer is right?

Because it was checked before you saw it. Every candidate antiderivative is differentiated and compared against what you typed, at several values, and anything that does not match is not shown. You can do the same check by hand, and on an exam you should: differentiating your answer is the fastest way to catch a slip.

Why does my answer look different from the calculator's?

Two antiderivatives can differ by a constant and still both be right, so your version and this one may simply have absorbed a different constant into C. They can also be algebraically different but equal: sin squared x over 2 and minus cos squared x over 2 differ by a constant, and the two halves of a partial fraction answer can be combined into a single logarithm. If you are unsure, differentiate both.

Can it do definite integrals?

No. This one finds antiderivatives. For a definite integral, take the antiderivative it gives you and evaluate it at the two limits, subtracting the lower from the upper — being careful that the function has no discontinuity between them, which is where that shortcut goes wrong.

Is anything I type stored or sent anywhere?

No. The parser, the integration techniques and the check that verifies the answer are all JavaScript running in your browser, with no request to any server. Nothing you type is stored, logged or sent, and the page works with your connection off once it has loaded.

Integration is where calculus gets hard

Differentiating is a procedure you can follow. Integrating is a search: the work is recognising which technique a problem wants, and that is what practice with someone watching actually builds. We tutor UBC and SFU integral calculus and BC Calculus 12 one-on-one, starting from the problem in front of you.

Get calculus helpBook a session

Derivative calculator

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